Monge-Ampère gravity: From the large deviation principle to cosmological simulations through optimal transport

We study Monge-Amp\`ere gravity (MAG) as an effective theory of cosmological structure formation through optimal transport theory. MAG is based on the Monge-Amp\`ere equation, a nonlinear version of the Poisson equation, that relates the Hessian determinant of the potential to the density field. We explain how MAG emerges from a conditioned system of independent and indistinguishable Brownian particles, through the large deviation principle, in the continuum limit. To numerically explore this highly nonlinear theory, we develop a novel $N$-body simulation method based on semidiscrete optimal transport. Our results obtained from the very first $N$-body simulation of Monge-Amp\`ere gravity with over 100 millions particles show that on large scales, Monge-Amp\`ere gravity is similar to the Newtonian gravity but favors the formation of anisotropic structures such as filaments. At small scales, MAG has a weaker clustering and is screened in high-density regions. Although here we study the Monge-Amp\`ere gravity as an effective rather than a fundamental theory, our novel highly performant optimal transport algorithm can be used to run high-resolution simulations of a large class of modified theories of gravity, such as Galileons, in which the equations of motion are second-order and of Monge-Amp\`ere type.